These London Airports are served from Edinburgh, by the airlines shown:
London City: CityJet & BA
Gatwick: easyJet & BA
Heathrow: Bmi & BA
Luton: easyJet
Stansted: easyJet
easyJet offer £40.99 outbound, as follows:
1905 to Stansted, landing at 2025
1955 to Luton, landing at 2110
2000 to Gatwick, landing 2130
2050 to Stansted, landing at 2210
2115 to Gatwick, landing at 2245
You've not stated what time you want to return on the 31st. easyJet have a total of 12 flights available, across the three airports, ranging from £34.99 (ex Stansted at 1140, arriving at 1255) up to £78.99.
So easyJet to & from Stansted might be your best option. (I've not checked out the other airlines but they won't be significantly cheaper).
Trains leave Stansted every 15 minutes, taking 35 minutes to get to Tottenham Hale. From there it's just 3 stops along the Victoria Line (under 10 minutes) to Highbury & Islington station.
Chris